Question

How do you find the derivative of $u = \sin \left( {{e^t}} \right)$ ?

Answer 1

Here we are asked to find the derivative of a given function. Derivative of a function can be found by using the standard formula that is available in differentiation but here we can see that the given function is a composition function that is the function inside a function. So first we will find the derivative of the outer function by substituting a variable as an inner function then we will find the derivative of the inner function then the product of those two will give the required derivative.

Complete step by step answer:
It is given that $u = \sin \left( {{e^t}} \right)$ we aim to find the derivative of the given function. As we can see that the given function is a composition function which is of the form $f\left( {g\left( x \right)} \right)$ where $f\left( x \right)$ and $g\left( x \right)$ are two separate functions.
Here $f\left( x \right) = \sin (x)$ where $x = {e^x}$ and $g\left( x \right) = {e^x}$ .
The derivative of a composition function can be done by the following
$\dfrac{d}{{dx}}\left( {f\left( {g\left( x \right)} \right)} \right) = f'\left( {g\left( x \right)} \right)g'\left( x \right)$
Then the derivative of the given composition function can be found by using the above before we need to give a substitution for the inner function.
Let $v = {e^t}$ so that we get
$\dfrac{d}{{dt}}\left( {\sin \left( {{e^t}} \right)} \right) = \dfrac{d}{{dv}}\left( {\sin v} \right)\dfrac{d}{{dt}}\left( {{e^t}} \right)$
Using the formula $\dfrac{d}{{dx}}\sin x = \cos x$ concerning the variable $v$ we get
$\Rightarrow \cos v\dfrac{d}{{dt}}\left( {{e^t}} \right)$
Now using the formula $\dfrac{d}{{dx}}{e^x} = {e^x}$ concerning the variable $t$ we get
$\Rightarrow \cos v.\left( {{e^t}} \right)$
Now let us re substitute the temporary variable that we substituted for the inner function that is $v = {e^t}$ .
$\Rightarrow {e^t}.\cos \left( {{e^t}} \right)$
Thus, we have found the derivative of the given function that is $\dfrac{d}{{dt}}\left( {\sin \left( {{e^t}} \right)} \right) = {e^t}.\cos \left( {{e^t}} \right)$ .

Note: In the above problem we have substituted a temporary variable to the inner function of the composition function, it is necessary to re-substitute them again to get the original variable that is given in the problem. The rule that is used to find the derivative of a composition function is called a chain rule that is finding derivatives for the outer function and then the inner function.